Dynamical Systems: Stability, Symbolic Dynamics, and Chaos 2nd Edition (Studies in Advanced Mathematics)

522 Pages·1998·15.986 MB·English

by  Clark Robinson

DYNAMICAL
SYSTEMS
Stability,
Symbolic Dynamics,
and Chaos
Second Edition
Clark Robinson
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business

DYNAMICAL
SYSTEMS
Stability,
Symbolic Dynamics,
and Chaos
Second Edition
Published in 1999 by
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
' 1999 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
15 14 13 12 11 10 9 8 7 6 5
International Standard Book Number-10: 0-8493-8495-8 (Hardcover)
International Standard Book Number-13: 978-0-8493-8495-0 (Hardcover)
Library of Congress catalog number: 98-34470
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Library of Congress Cataloging-in-Publication Data
Robinson, C. (Clark)
Dynamical systems : stability, symbolic dynamics, and chaos / Clark Robinson.(cid:151)[2nd ed.]
p. cm. (cid:151) (Studies in Advanced Mathematics)
Includes bibliographical references (p. - ) and index
ISBN 0-8493-8495-8 (alk. paper)
1. Differentiable dynamical systems. I. Title. II. Series.
QA614.8.R63 1998
515’.352(cid:151)dc21 98-34470
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Studies in Advanced Mathematics
Titles Included in the Series
John P. D’Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces
Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping
John J. Benedetto, Harmonic Analysis and Applications
John J. Benedetto and Michael W. Frazier, Wavelets: Mathematics and Applications
Albert Boggess, CR Manifolds and the Tangential Cauchy–Riemann Complex
Keith Burns and Marian Gidea, Differential Geometry and Topology: With a View to Dynamical Systems
George Cain and Gunter H. Meyer, Separation of Variables for Partial Differential Equations: An
Eigenfunction Approach
Goong Chen and Jianxin Zhou, Vibration and Damping in Distributed Systems
Vol. 1: Analysis, Estimation, Attenuation, and Design
Vol. 2: WKB and Wave Methods, Visualization, and Experimentation
Carl C. Cowen and Barbara D. MacCluer, Composition Operators on Spaces of Analytic Functions
Jewgeni H. Dshalalow, Real Analysis: An Introduction to the Theory of Real Functions and Integration
Dean G. Duffy, Advanced Engineering Mathematics with MATLAB®, 2nd Edition
Dean G. Duffy, Green’s Functions with Applications
Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of Functions
Gerald B. Folland, A Course in Abstract Harmonic Analysis
José García-Cuerva, Eugenio Hernández, Fernando Soria, and José-Luis Torrea,
Fourier Analysis and Partial Differential Equations
Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem,
2nd Edition
Peter B. Gilkey, John V. Leahy, and Jeonghueong Park, Spectral Geometry, Riemannian Submersions,
and the Gromov-Lawson Conjecture
Alfred Gray, Elsa Abbena, and Simon Salamon Modern Differential Geometry of Curves and Surfaces
with Mathematica, Third Edition
Eugenio Hernández and Guido Weiss, A First Course on Wavelets
Kenneth B. Howell, Principles of Fourier Analysis
Steven G. Krantz, The Elements of Advanced Mathematics, Second Edition
Steven G. Krantz, Partial Differential Equations and Complex Analysis
Steven G. Krantz, Real Analysis and Foundations, Second Edition
Kenneth L. Kuttler, Modern Analysis
Michael Pedersen, Functional Analysis in Applied Mathematics and Engineering
Clark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition
John Ryan, Clifford Algebras in Analysis and Related Topics
John Scherk, Algebra: A Computational Introduction
Jonathan D. H. Smith, Quasigroup Representation Theory
Pavel Sˇolín, Karel Segeth, and Ivo Doleˇzel, High-Order Finite Element Method
André Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones
James S. Walker, Fast Fourier Transforms, 2nd Edition
James S. Walker, A Primer on Wavelets and Their Scientific Applications
Gilbert G. Walter and Xiaoping Shen, Wavelets and Other Orthogonal Systems, Second Edition
Nik Weaver, Mathematical Quantization
Kehe Zhu, An Introduction to Operator Algebras
Preface to the First Edition
In recent years, Dynamical Systems has had many applications in science and engi-
neering,someofwhichhavegoneundertherelatedheadingsofchaostheoryornonlinear
analysis. Behindtheseapplicationsthereliesarichmathematicalsubjectwhichwetreat
in this book. This subject centers on the orbits of iteration of a (nonlinear) function
or of the solutions of (nonlinear) ordinary differential equations. In particular, we are
interested in the properties which persist under nonlinear change of coordinates. As
such, we areinterested in the geometric or topologicalaspects ofthe orbits or solutions
more than an explicit formula for an orbit (which may not be available in any case).
However, as becomes clear in the treatment in this book, there are many properties
of a particular solution or the whole system which can be measured by some quantity.
Also, althoughthe subject has a geometric ortopologicalflavor,analytic analysisplays
an important role (e.g., the local analysis near a fixed point and the stable manifold
theory).
TherehavebeenseveralbooksandmonographsonthesubjectofDynamicalSystems.
There are severaldistinctive aspects which together make this book unique.
First of all, this book treats the subject from a mathematical perspective with the
proofs of most of the results included: the only proofs which are omitted either (i) are
left to the reader, (ii) are too technically difficult to include in an introductory book,
even at the graduate level, or (iii) concern a topic which is only included as a bridge
between the material covered in the book and commonly encountered concepts in Dy-
namical Systems. (Much of the material concerning measures, Liapunov exponents,
and fractal dimension is of this latter category.) Although it has a mathematical per-
spective, readers who are more interested in applied or computational aspects of the
subject should find the explicit statements of the results helpful even if they do not
concernthemselveswiththedetailsofthe proofs. Inparticular,theinclusionofexplicit
formulas for the various bifurcations should be very useful.
Second, this book is meant to be a graduate textbook and not just a reference book
or monograph on the subject. This aspect of the book is reflected in the way the
background materials are carefully reviewed as we use them. (The particular prerequi-
sites from undergraduate mathematics are discussed below.) The ideas are introduced
through examples and at a level which is accessible to a beginning graduate student.
Many exercisesare included to help the student learnthe meaning of the theorems and
master the techniques of the proofs and topic under consideration. Since the exercises
are not usually just routine applications of theorems but involve similar proofs and or
calculations,they arebestassignedingroups,weeklyorbiweekly. Forthis reason,they
are grouped at the end of each chapter rather than in the individual section.
Third, the scope of the book is on the scale of a year long graduate course and is
designedtobeusedinsuchagraduatelevelmathematicscourseinDynamicalSystems.
Thismeansthatthebookisnotcomprehensiveorexhaustivebuttriestotreatthecore
concepts thoroughly and treat others enough so the reader will be prepared to read
further in Dynamical Systems without a complete mathematical treatment. In fact,
this bookgrewoutofagraduatecoursethatItaughtatNorthwesternUniversitymany
times between the early 1970s and the present. To the material that I covered in that
course, I have added a few other topics: some of which my colleagues treat when they
teach the course, others round out the treatment of a topic covered earlier in the book
(e.g., Chapter XII∗), and others just give greater flexibility to possible courses using
this book. Details on which sections form the core of the book are discussed in Section
1.4.
Theperspectiveofthebookiscenteredonmultidimensionalsystemsofrealvariables.
Chapters II and III concern functions of one real variable, but this is done mainly
because this makes the treatment simpler analytically than that given later in higher
dimensions: there are not any (or many) aspects introduced which are unique to one
dimension. Some results are proved so they apply in Banach spaces or even complete
metricbutmostoftheresultsaredevelopedinfinitedimensions. Inparticular,nodirect
connectionwith partialdifferential equations or delay equations is given. The fact that
the book concerns functions of real rather than complex variables explains why topics
such as the Julia set, Mandelbrot set, and Measurable Riemann Mapping Theorem are
not treated.
Thisbooktreatsthedynamicsofbothiterationoffunctionsandsolutionsofordinary
differentialequations. Manyoftheconceptsarefirstintroducedforiterationoffunctions
wherethe geometryis simpler,butanattempt hasbeenmade to interpretthese results
for differential equations. A proof of the existence and continuity of solutions with
respectto initialconditions is alsoincluded to establishthe beginnings ofthis aspectof
the subject.
Although there is much overlap in this book and one on ordinary differential equa-
tions, the emphasis is different. The dynamical systems approach centers more on
properties of the whole system or subsets of the system rather than individual solu-
tions. Even the more local theory in Chapters IV–VII deals with characterizing types
of solutions under various hypotheses. Chapters VIII and X deal more directly with
more global aspects: Chapter VIII centers on various examples and Chapter X gives
the global theory.
Finally, within the varioustypes of DynamicalSystems, this book is mostconcerned
withhyperbolicsystems: thisfocusismostprominentinChaptersVIII,X,XI,andXII.
However,anattempt has been made to makethis book valuable to people interestedin
various aspects of Dynamical Systems.
The specific prerequisites include undergraduate analysis (including the Implicit
FunctionTheorem),linearalgebra(includingthe Jordancanonicalform),andpointset
topology (including Cantor sets). For the analysis, one of the following books should
be sufficient background: Apostol (1974), Marsden (1974), or Rudin (1964). For the
linearalgebra,oneofthefollowingbooksshouldbesufficientbackground: Hoffmanand
Kunze (1961) or Hartley and Hawkes (1970). For the point set topology, one of the
following books should be sufficient background: Croom (1989), Hocking and Young
(1961), or Munkres (1975). What is needed from these other subjects is an ability to
usethesetools;knowingaproofoftheImplicitFunctionTheoremdoesnotparticularly
help someone know how to use it. For this reason, we carefully discuss the way these
tools are used just before we use them. (See the sections on the Calculus Prerequisites,
Cantor Sets, Real Jordan Canonical Form, Differentiation in Higher Dimensions, Im-
plicit Function Theorem, Inverse Function Theorem, Contraction Mapping Theorem,
andDefinitionofaManifold.) AfterusingthesetoolsinDynamicalSystems,thereader
should gain a much better understanding of the importance of these “undergraduate”
subjects. The terminology and ideas from differential topology or differential geometry
*Chapter numbershavebeenrevisedtoagreewiththoseinthecurrentedition.
are also used, including that of a tangent vector, the tangent bundle, and a mani-
fold. However, most surfaces or manifolds are either Euclidean space, tori, or graphs
of functions so these ideas should not be too intimidating. Although someone pursuing
DynamicalSystemsfurthershouldlearnmanifoldtheory,Ihavetriedtomakethisbook
accessibletosomeonewithoutpriorbackgroundinthis subject. Thus,the prerequisites
for this book are really undergraduate analysis, linear algebra, and point set topology
and not advanced graduate work. However, the reader should be warned that most
beginning graduate students do not find the material at all trivial. The main compli-
cating aspect seems to be the use of a large variety of methods and approaches. The
unifying feature is not the methods used but the type of questions which we are trying
to answer. By having patience and reviewing the mathematics from other subjects as
they are used, the reader should find the material accessible and rich in content, both
mathematical and for applications.
The main topic of the book is the dynamics induced by iteration of a (nonlinear)
function or by the solutions of (nonlinear) ordinary differential equations. In the usual
undergraduate mathematics courses, some properties of solutions of differential equa-
tions are considered but more attention is paid to the specific form of the solution. In
connection with functions, they are graphed and their minima and maxima are found,
but the iterates of a function are not often considered. To iterate a function we repeat-
edly have the same function act on a point and its images. Thus, for a function f with
initial condition x0, we consider x1 =f(x0), and then xn =f(xn−1) for n≥1. We are
interestedinfindingthequalitativefeaturesandlongtimelimitingbehaviorofatypical
orbit,foreitheranordinarydifferentialequationortheiteratesofafunction. Certainly,
fixed points or periodic points are important, but sometimes the orbit moves densely
through a complicated set such as a Cantor set. We want to understand and bring a
structure to this seemingly randombehavior. It is often expressed by saying,“we want
tobringorderoutofchaos.” Onewayoffindingthisstructureisviathetoolofsymbolic
dynamics. If there is a real valued function f and a sequence of intervals Ji such that
the image of Ji by f covers Ji+1, f(Ji) ⊃ Ji+1, then it is possible to show that there
is a point x whose orbit passes through this sequence of intervals, fi(x) ∈ Ji. Labels
forthe intervalsthencanbe usedas symbols,hence the name ofsymbolicdynamics for
this approach.
Another important conceptis that of structuralstability. Some types ofsystems (it-
eratedfunctions or ordinarydifferentialequations)have dynamics which areequivalent
(topologically conjugate) to that of any of its perturbations. Such a system is called
structurally stable. Finally, the term chaos is given a special meaning and interpreta-
tion. There is no one set definition of a chaotic system, but we discuss various ideas
and measurements related to chaotic dynamics. One of the ironies is that some chaotic
systems are also structurally stable.
Chapter I gives a more detailed introduction into the main ideas that are treated in
the book by means of examples of functions and differential equations. Suffice it to say
herethatthesethreeideas,symbolicdynamics,structuralstability,andchaos,formthe
central part of the approach to Dynamical Systems presented in this book.
Intheyear-longgraduatecourseatNorthwestern,wecoverthethematerialinChap-
ter II, Sharkovskii’s Theorem and Subshifts of Finite Type from Chapter III, Chapter
IV except the Perron-Frobenius Theorem, Chapter V except some of the material on
periodic orbits for planar differential equations (and sometime the proof of the Stable
Manifold Theorem is omitted), a selection of examples from Chapter VIII, and most
of Chapter X. In a given year, other selected topics are usually added from among the
following: Chapter VII on bifurcations, the material on topologicalentropy in Chapter
IX,andtheKupka-SmaleTheorem. Acoursewhichdidnotemphasizetheglobalhyper-
bolic theory as much could be obtained by skipping Chapter X and treating additional
topics, e.g., Chapter VII or more on the measurements of chaos. Section 1.4 discusses
the content of the different chapters and possible selections of sections or topics for a
course using this book.
There are several other books which give introductions into other aspects or ap-
proaches to Dynamical Systems. For other graduate level mathematical introductions
to DynamicalSystems, see Devaney (1989),Irwin(1980),Nitecki(1970),andPalisand
de Melo(1982). ForamorecomprehensivetreatmentofDynamicalSystems,seeKatok
and Hasselblatt (1994). Some books which emphasize the dynamics of iteration of a
function of one variable are Alseda`, Llibre, and Misiurewicz (1993), Block and Coppel
(1992), and de Melo and Van Strien (1993). Carleson and Gamelin (1993) gives an
introductionto thedynamicsoffunctions ofacomplexvariable. ChowandHale(1982)
givesa morethoroughtreatmentofthe bifurcationaspects ofDynamicalSystems. The
articlebyBoyle(1993)givesamorethoroughintroductionintosymbolicdynamicsasa
separatesubjectandnotjusthowitisusedtoanalyzediffeomorphismsorvectorfields.
Some books which concentrate on Hamiltonian dynamics are Abraham and Marsden
(1978), Arnold (1978), and Meyer and Hall (1992). For an introduction to applications
ofDynamicalSystems,seeGuckenheimerandHolmes(1983),HirschandSmale(1974),
Wiggins (1990, 1988), and Ott (1993). For applications to ecology, see Hirsch (1982,
1985, 1988, 1990), Hofbauer and Sigmund (1988), Hoppensteadt (1982), May (1975),
and Waltman (1983). There are many books written on Dynamical Systems by people
in fields outside mathematics, including LichtenbergandLieberman(1983),Marekand
Schreiber (1991), and Rasband (1990).
I have tried very hard to give references to original papers. However, there are
many researchers working in Dynamical Systems and I am not always aware of (or
remember) contributions by various people to which I should give credit. I apologize
formyomissions. Iamsuretherearemany. Ihope the referencesthatIhavegivenwill
help the reader start finding the related work in the literature.
When referring to a theorem in the same chapter, we use the number as it appears
in the statement, e.g., Theorem 2.2 which is the second theorem of the second section
ofthecurrentchapter. IfwearereferringtoTheorem2.2fromChapterVI inachapter
otherotherthanChapterVI,werefertoitasTheoremVI.2.2toindicateitcomesfrom
a different chapter.
There are not any specific references in this book to using a computer to simulate a
dynamicalsystem. However,thereaderwouldbenefitgreatlybyseeingthedynamicsas
it unfolds by such simulation. The reader can either write a programfor him or herself
or use several of the computer packages available. On an IBM Personal Computer,
I have used the program Phaser which comes with the book by Koc¸ak (1989). The
program Dynamics by Nusse and Yorke (1990) runs on both IBM Personal Computers
andUnix/X11machines. ThereareseveralotherprogramsforIBMPersonalComputers
but I have not used them myself. Also, the program DSTool by J. Guckenheimer, M.
R. Myers, F. J. Wicklin, and P. A. Worfolk runs on Unix/X11 machines. Many of the
programminglanguagescomewithagoodenoughgraphicslibrarythatitisnotdifficult
to write one’s own specialized program. However,for the X-Window environment on a
Unixcomputer,IfoundtheVOGLE library(CgraphicsCfunctions)averyhelpfulasset
to write my own programs. There are several programs for the Macintosh, including
MacMath by Hubbard and West (1992), but I have not used them.
Overtheyears,IhavehadmanyusefulconversationswithcolleaguesatNorthwestern
University and from elsewhere, especially people attending the Midwest Dynamical
Systems Seminars. Those colleaguesin Dynamical Systems at NorthwesternUniversity
includeKeithBurns,JohnFranks,DonSaari,RobertWilliams,andmanypostdoctoral